#8505963

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,4)(5,6)(7,8)(9,10,14,18)(11,15,21,16)(12,17,23,13)(19,20,22,24)(28,36)(32,37)(35,40)(38,39);(2,6)(3,7)(9,11,14,21)(10,15,18,16)(12,22,23,19)(13,20,17,24)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,38)(39,40);(1,3,5,7)(2,4,6,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,19)(13,24)(14,21)(16,18)(17,20)(22,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(9,12)(10,17)(11,19)(13,18)(14,23)(15,20)(16,24)(21,22)(25,28,30,36)(26,32,34,37)(27,35,33,40)(29,38,31,39);(9,13,14,17)(10,12,18,23)(11,20,21,24)(15,22,16,19)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,40,37,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413159);